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<meta name="description" content="处理方式 将原始矩阵正向化 正向矩阵标准化 计算得分并归一化  正向化矩阵正向化就是要将所有的指标类型统一转化为极大型指标。 常见的四种指标：     指标名称 指标特点 例子     极大型指标 越大越好 成绩、GDP增速、企业利润   极小型指标 越小越好 费用、坏品率、污染程度   中间型指标 越接近某个值越好 PH值   区间型指标 落在某个区间最好 体温、营养物量     正向化公式极小">
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<meta name="twitter:description" content="处理方式 将原始矩阵正向化 正向矩阵标准化 计算得分并归一化  正向化矩阵正向化就是要将所有的指标类型统一转化为极大型指标。 常见的四种指标：     指标名称 指标特点 例子     极大型指标 越大越好 成绩、GDP增速、企业利润   极小型指标 越小越好 费用、坏品率、污染程度   中间型指标 越接近某个值越好 PH值   区间型指标 落在某个区间最好 体温、营养物量     正向化公式极小">
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        <h2 id="处理方式"><a href="#处理方式" class="headerlink" title="处理方式"></a>处理方式</h2><ol>
<li>将原始矩阵正向化</li>
<li>正向矩阵标准化</li>
<li>计算得分并归一化</li>
</ol>
<h3 id="正向化"><a href="#正向化" class="headerlink" title="正向化"></a>正向化</h3><p>矩阵正向化就是要将所有的指标类型统一转化为极大型指标。</p>
<p>常见的四种指标：</p>
<div class="table-container">
<table>
<thead>
<tr>
<th>指标名称</th>
<th>指标特点</th>
<th>例子</th>
</tr>
</thead>
<tbody>
<tr>
<td>极大型指标</td>
<td>越大越好</td>
<td>成绩、GDP增速、企业利润</td>
</tr>
<tr>
<td>极小型指标</td>
<td>越小越好</td>
<td>费用、坏品率、污染程度</td>
</tr>
<tr>
<td>中间型指标</td>
<td>越接近某个值越好</td>
<td>PH值</td>
</tr>
<tr>
<td>区间型指标</td>
<td>落在某个区间最好</td>
<td>体温、营养物量</td>
</tr>
</tbody>
</table>
</div>
<h3 id="正向化公式"><a href="#正向化公式" class="headerlink" title="正向化公式"></a>正向化公式</h3><h4 id="极小型"><a href="#极小型" class="headerlink" title="极小型"></a>极小型</h4><script type="math/tex; mode=display">
\hat{x_i}=max(x_i)-x_i</script><h4 id="中间型"><a href="#中间型" class="headerlink" title="中间型"></a>中间型</h4><script type="math/tex; mode=display">
M=max{|x_i-x_{best}|},\hat{x_i}=1-\frac{|x_i-x_{best}|}{M}</script><h4 id="区间型"><a href="#区间型" class="headerlink" title="区间型"></a>区间型</h4><script type="math/tex; mode=display">
M=max\{a-min\{x_i\},max\{x_i\}-b\},\\
\hat{x_i}=\begin{cases}
    1-\frac{a-x}{M},x<a\\
    1       ,a\leq x \leq b\\
    1-\frac{x-b}{M},x>b
\end{cases}</script><h3 id="正向化矩阵标准化"><a href="#正向化矩阵标准化" class="headerlink" title="正向化矩阵标准化"></a>正向化矩阵标准化</h3><script type="math/tex; mode=display">
每个元素/\sqrt{其所在列的元素的平方和}</script><h3 id="计算得分并归一化"><a href="#计算得分并归一化" class="headerlink" title="计算得分并归一化"></a>计算得分并归一化</h3><p><img src="http://m.qpic.cn/psc?/V11NehB63qJi50/xZikVHqhLrt9jsfqm9tF*Qh93FBXh*GEN2j3mjdM3b6eO1sgHCoODTzD6Kuz4kQLBl66VHXe7VE2RGlxBUr7IA!!/b&bo=AQRrAgAAAAARB1w!&rf=viewer_4"></p>
<h2 id="示例代码"><a href="#示例代码" class="headerlink" title="示例代码"></a>示例代码</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">%正向化处理</span></span><br><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[posit_x]</span> = <span class="title">Positivization</span><span class="params">(x,type,i)</span></span></span><br><span class="line">    <span class="keyword">if</span> <span class="built_in">type</span> == <span class="number">1</span>  <span class="comment">%极小型</span></span><br><span class="line"></span><br><span class="line">        posit_x = Min2Max(x);  <span class="comment">%调用Min2Max函数来正向化</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">elseif</span> <span class="built_in">type</span> == <span class="number">2</span>  <span class="comment">%中间型</span></span><br><span class="line"></span><br><span class="line">        best = input(<span class="string">'请输入最佳的那一个值： '</span>);</span><br><span class="line">        posit_x = Mid2Max(x,best);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">elseif</span> <span class="built_in">type</span> == <span class="number">3</span>  <span class="comment">%区间型</span></span><br><span class="line"></span><br><span class="line">        a = input(<span class="string">'请输入区间的下界： '</span>);</span><br><span class="line">        b = input(<span class="string">'请输入区间的上界： '</span>); </span><br><span class="line">        posit_x = Inter2Max(x,a,b);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 极小型</span></span><br><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[posit_x]</span> = <span class="title">Min2Max</span><span class="params">(x)</span></span></span><br><span class="line">    posit_x = <span class="built_in">max</span>(x) - x;</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 中间型</span></span><br><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[posit_x]</span> = <span class="title">Mid2Max</span><span class="params">(x,best)</span></span></span><br><span class="line">    M = <span class="built_in">max</span>(<span class="built_in">abs</span>(x-best));</span><br><span class="line">    posit_x = <span class="number">1</span> - <span class="built_in">abs</span>(x-best) / M;</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 区间型</span></span><br><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[posit_x]</span> = <span class="title">Inter2Max</span><span class="params">(x,a,b)</span></span></span><br><span class="line">    r_x = <span class="built_in">size</span>(x,<span class="number">1</span>);  </span><br><span class="line">    M = <span class="built_in">max</span>([a-<span class="built_in">min</span>(x),<span class="built_in">max</span>(x)-b]);</span><br><span class="line">    posit_x = <span class="built_in">zeros</span>(r_x,<span class="number">1</span>);   </span><br><span class="line">    <span class="keyword">for</span> <span class="built_in">i</span> = <span class="number">1</span>: r_x</span><br><span class="line">        <span class="keyword">if</span> x(<span class="built_in">i</span>) &lt; a</span><br><span class="line">           posit_x(<span class="built_in">i</span>) = <span class="number">1</span>-(a-x(<span class="built_in">i</span>))/M;</span><br><span class="line">        <span class="keyword">elseif</span> x(<span class="built_in">i</span>) &gt; b</span><br><span class="line">           posit_x(<span class="built_in">i</span>) = <span class="number">1</span>-(x(<span class="built_in">i</span>)-b)/M;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">           posit_x(<span class="built_in">i</span>) = <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">end</span></span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line">clear;clc</span><br><span class="line"></span><br><span class="line">[n,m] = <span class="built_in">size</span>(X);</span><br><span class="line"><span class="built_in">disp</span>([<span class="string">'共有'</span> num2str(n) <span class="string">'个评价对象, '</span> num2str(m) <span class="string">'个评价指标'</span>]) </span><br><span class="line">Judge = input([<span class="string">'这'</span> num2str(m) <span class="string">'个指标是否需要经过正向化处理，需要请输入1 ，不需要输入0：  '</span>]);</span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> Judge == <span class="number">1</span></span><br><span class="line">    Position = input(<span class="string">'请输入需要正向化处理的指标所在的列，例如第2、3、6三列需要处理，那么你需要输入[2,3,6]： '</span>); <span class="comment">%[2,3,4]</span></span><br><span class="line">    <span class="built_in">disp</span>(<span class="string">'请输入需要处理的这些列的指标类型（1：极小型， 2：中间型， 3：区间型） '</span>)</span><br><span class="line">    Type = input(<span class="string">'例如：第2列是极小型，第3列是区间型，第6列是中间型，就输入[1,3,2]：  '</span>); </span><br><span class="line">    <span class="comment">% 注意，Position和Type是两个同维度的行向量</span></span><br><span class="line">    <span class="keyword">for</span> <span class="built_in">i</span> = <span class="number">1</span> : <span class="built_in">size</span>(Position,<span class="number">2</span>)  <span class="comment">%这里需要对这些列分别处理，因此我们需要知道一共要处理的次数，即循环的次数</span></span><br><span class="line">        X(:,Position(<span class="built_in">i</span>)) = Positivization(X(:,Position(<span class="built_in">i</span>)),Type(<span class="built_in">i</span>),Position(<span class="built_in">i</span>));</span><br><span class="line">    <span class="comment">% Positivization是我们自己定义的函数，其作用是进行正向化，其一共接收三个参数</span></span><br><span class="line">    <span class="comment">% 第一个参数是要正向化处理的那一列向量 X(:,Position(i))   回顾上一讲的知识，X(:,n)表示取第n列的全部元素</span></span><br><span class="line">    <span class="comment">% 第二个参数是对应的这一列的指标类型（1：极小型， 2：中间型， 3：区间型）</span></span><br><span class="line">    <span class="comment">% 第三个参数是告诉函数我们正在处理的是原始矩阵中的哪一列</span></span><br><span class="line">    <span class="comment">% 该函数有一个返回值，它返回正向化之后的指标，我们可以将其直接赋值给我们原始要处理的那一列向量</span></span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line">    <span class="built_in">disp</span>(<span class="string">'正向化后的矩阵 X =  '</span>)</span><br><span class="line">    <span class="built_in">disp</span>(X)</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line">Z = X ./ <span class="built_in">repmat</span>(sum(X.*X) .^ <span class="number">0.5</span>, n, <span class="number">1</span>);</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'标准化矩阵 Z = '</span>)</span><br><span class="line"><span class="built_in">disp</span>(Z)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">D_P = sum([(Z - <span class="built_in">repmat</span>(<span class="built_in">max</span>(Z),n,<span class="number">1</span>)) .^ <span class="number">2</span> ],<span class="number">2</span>) .^ <span class="number">0.5</span>;   <span class="comment">% D+ 与最大值的距离向量</span></span><br><span class="line">D_N = sum([(Z - <span class="built_in">repmat</span>(<span class="built_in">min</span>(Z),n,<span class="number">1</span>)) .^ <span class="number">2</span> ],<span class="number">2</span>) .^ <span class="number">0.5</span>;   <span class="comment">% D- 与最小值的距离向量</span></span><br><span class="line">S = D_N ./ (D_P+D_N);    <span class="comment">% 未归一化的得分</span></span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'最后的得分为：'</span>)</span><br><span class="line">stand_S = S / sum(S)</span><br><span class="line">[sorted_S,index] = <span class="built_in">sort</span>(stand_S ,<span class="string">'descend'</span>)</span><br></pre></td></tr></table></figure>

      
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#处理方式"><span class="nav-number">1.</span> <span class="nav-text">处理方式</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#正向化"><span class="nav-number">1.1.</span> <span class="nav-text">正向化</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#正向化公式"><span class="nav-number">1.2.</span> <span class="nav-text">正向化公式</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#极小型"><span class="nav-number">1.2.1.</span> <span class="nav-text">极小型</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#中间型"><span class="nav-number">1.2.2.</span> <span class="nav-text">中间型</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#区间型"><span class="nav-number">1.2.3.</span> <span class="nav-text">区间型</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#正向化矩阵标准化"><span class="nav-number">1.3.</span> <span class="nav-text">正向化矩阵标准化</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#计算得分并归一化"><span class="nav-number">1.4.</span> <span class="nav-text">计算得分并归一化</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#示例代码"><span class="nav-number">2.</span> <span class="nav-text">示例代码</span></a></li></ol></div>
            

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